Research On GPU Acceleration Of The Coupled FE-EFG Method For Parallel Computing

Element Free Galerkin Method(EFGM) and Finite Element Method(FEM) have their own advantages and disadvantages. EFGM gets rid of the shackles of the grid, has the high degree of accuracy and is prevalent on discontinuous problems, but it is inconvenient for dealing with boundary conditions and low-efficient in computing. FEM has the high computing efficiency, but it meets difficulties when dealing with problems such as large deformation, crack progagations and topology optimization which can easily product grid distortion. The coupled Finite Element-Element free Galerkin(FE-EFG) Method combines the advantages of FEM and EFGM respectively. It can improve the degree of accuracy and simplify the process of solving certain problem with the proper meshing. Nevertheless, the computational cost is still expensive with the use of coupled method, which has limited the application in large-scale engineering problems. In the past dozen years, Graphic Processing Unit(GPU) parallel computing provided a satisfactory way to solve the inefficiency problem, which is rapidly developed in high performance computing field. This paper mainly researches the coupled method based on GPU parallel computing for reducing the computing time. The main research contents are summarized as follows:(1) The coupled FE-EFG method based on the collocation approach is described and examined in the paper. Displacement boundary conditions are applied to finite element by standard methods, and the unified form has been derived for imposing the given and symmetric displacement boundary conditions. The conventional coupled FE-EFG method has been carried on the implementation of the numerical test and shows that the coupled method gives the reasonably accurate results.(2) An improved algorithm of the coupled method is introduced, which is based on Compress Sparse Row(CSR) compressed storage format and the high efficient iterative solvers. The main improved point is assembling the overall stiffness matrix through the unit of non-zero sub-block stiffness matrix by node pairs instead of integral points or FE cells. The method of finding the node pairs and the formula of calculating the sub-block stiffness matrix are given. The high efficient iterative algorithms, Conjungate Gradient(CG) and Preconditioned Conjungate Gradient(PCG), are applied to the coupled method in this paper. The numercical test shows that the improved algorithm gives a correct solution. The analysis and comparison between the CG and PCG solutions demonstrate that the PCG has minor costs in accuracy degradation, but its efficiency is much higher than CG method.(3) According to the improved algorithm of the coupled FE-EFG method, the GPU parallel computing algorithm is studied. With analysising the time consuming of each part of the improved coupling algorithm, the paper proposed an efficient scheme that makes the stiffness matrix of the FE field including the interface part calculated in Graphic Processing Unit(CPU) and the EFG part in Graphic Processing Unit(GPU). And through using CUBLAS library and CUSPARSE library provided by Compute Unified Device Architecture(CUDA) platform, the overall discrete equation can be easily solved by the PCG algrithm. Meanwhile, GPU acceleration parallel program of the coupled FE-EFG method is written by CUDA C language. And the feasibility and correctness of the coupling FE-EFG algorithm based on GPU acceleration parallel were proved by the numerical examples, and.about 7 times speed up was achieved. In the end, the factors affecting speedup is discussed in this paper.The computing time of the coupled method is reduced significantly by GPU parallel computing. This research has an important theoretical reference value and significant with the use of the coupled method in large-scale engineering problem.